Free CAS Exam 5 Formula Sheet PDF, No Signup (2026)

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All Exam 5 Formulas

Ratemaking Data & Exposures 17 items

Calendar year incurred loss

L_{CY} = \text{Paid}_{\text{year}} + (R_{\text{end}} - R_{\text{begin}})

, Paid = losses paid during the year,

R_{end}

= ending case reserve,

R_{begin}

= beginning case reserve

Customer lifetime value closed form

\text{CLV} = \frac{m}{1 - r\,v}

, m = constant annual margin per policy, r = retention rate, v = annual discount factor

Transition-year claims-made rate from the cumulative report-lag step factor

\text{Rate}_n = \left(\sum_{j=0}^{n-1} p_j\right) \times \text{Mature Rate}

p_j

= share of ultimate cost at report lag j, n = policy maturity year

Excess ratio for large losses

\text{Excess Ratio} = \frac{\sum \text{losses above threshold}}{\sum \text{losses capped at threshold}}

, threshold = large-loss cap; loaded losses = capped losses

\times (1+\text{Excess Ratio})

Additive expense fee (policy fee) under the variable permissible loss ratio

\text{Fee} = \frac{F}{1 - V - Q}

, F = average fixed expense per exposure, V = variable expense %, Q = target profit %, denominator = variable permissible loss ratio

Off-balance factor for class relativity changes

OB = \dfrac{1}{1 + \Delta d}

, OB = off-balance factor applied to the base rate,

\Delta d

= change in the exposure-weighted average rate differential

Classical partial credibility square-root rule

Z = \sqrt{\frac{n}{n_{\text{full}}}}

, Z = credibility weight, n = observed claim count,

n_{\text{full}}

= full-credibility standard (e.g. 1,082 claims)

Exponentially trended value over a trend period

V_{\text{future}} = V_{\text{base}}\,(1+r)^{t}

V_{base}

= base value, r = annual trend rate, t = trend period in years (midpoint to midpoint)

Parallelogram on-level factor

F = \dfrac{I_{cur}}{\bar{I}}

, F = on-level factor,

I_{cur}

= current cumulative rate index,

\bar{I}

= area-weighted average rate index for the experience period

Bühlmann credibility factor

Z = \frac{n}{n + K}, \quad K = \frac{\text{EPV}}{\text{VHM}}

, n = claim count, K = Bühlmann constant, EPV = expected process variance, VHM = variance of hypothetical means

Frequency-severity pure premium cross check

\frac{C}{E} \times \frac{L}{C} = \frac{L}{E}

, C = claim count, E = earned exposures, L = incurred losses; frequency times severity must equal losses over exposures

Indicated rate change factor under the loss ratio method

\text{CF} = \dfrac{(L+E_L)/P + F}{1 - V - Q}

, L = losses,

E_L

= LAE, P = on-level premium, F = fixed expense ratio, V = variable expense ratio, Q = profit provision; percent change = CF − 1

Actuarially sound rate satisfying Principle 2 (all costs)

R = \dfrac{PP + F}{1 - V - Q}

, PP = pure premium (loss+LAE), F = fixed expense per exposure, V = variable expense ratio, Q = profit and contingency provision

Exposure trend factor for an inflation-sensitive base

(1 + g)^n

, g = annual exposure inflation rate (wage or price growth), n = trend period in years from historical midpoint to future policy midpoint

Indicated rate under the pure premium method

R = \frac{PP + F}{1 - V - Q}

, R = indicated rate per exposure, PP = pure premium (losses+LAE per exposure), F = fixed expense per exposure, V = variable expense %, Q = profit and contingencies %

GLM factor relativity under a log link

R_j = e^{\beta_j}

R_j

= multiplicative relativity for the factor level,

\beta_j

= fitted log relativity (coefficient); base level has

\beta_j=0

R_j=1.00

Net cost of reinsurance load

NC = \text{Ceded Prem} - E[\text{Recoveries}]

, NC = net cost loaded per exposure, Ceded Prem = reinsurance premium ceded, E[Recoveries] = expected ceded recoveries

Topic 0 18 items

Expected emergence over an interim period

E_{\text{emerge}} = \frac{U - A_{t_1}}{1 - \%_{t_1}} \times (\%_{t_2} - \%_{t_1})

, U = prior ultimate,

A_{t_1}

= actual emerged to date,

\%_{t_1}

= percent emerged at last valuation,

\%_{t_2}

= percent emerged at interim

Net unpaid claims from gross unpaid and unpaid recoveries

\text{Net Unpaid} = \text{Gross Unpaid} - \text{Unpaid S\&S} - \text{Unpaid Reins}

, S&S = salvage & subrogation, Reins = reinsurance recoverable, all unpaid = ultimate minus received to date

On-level severity restated to a common cost level via an inflation index

S_{\text{on-level}} = S_{\text{raw}} \times \frac{I_{\text{current}}}{I_{\text{claim year}}}

S_{raw}

= raw severity,

I_{current}

= index at target level,

I_{claim}

year = index at claim's cost level

Unpaid claim reserve from ultimate losses and percent paid

R = U \times (1 - p)

, R = unpaid claim reserve, U = ultimate losses, p = proportion of ultimate losses paid to date

Bornhuetter-Ferguson ultimate losses

U_{BF} = \text{Reported} + \text{Premium} \times \text{ELR} \times (1 - p)

, ELR = a priori expected loss ratio, p = 1/CDF = reported percentage, 1−p = unreported percentage

Unpaid ULAE under the classical paid-to-paid technique

\text{Unpaid ULAE} = R \times [0.50 \times C + 1.00 \times I]

, R = paid-ULAE-to-paid-loss ratio, C = case reserves, I = pure IBNR (claims not yet opened)

Total ultimate losses with a separate large-claim provision

U_{total} = L_{ult} + P_{large}

U_{total}

= total ultimate,

L_{ult}

= developed limited ultimate (limited losses × limited LDF),

P_{large}

= separately estimated large-claim provision

Berquist-Sherman adjusted reported losses for a cell

R_{adj} = \bar{C}_{adj}\,(N - K_{adj}) + P_{adj}

\bar{C}_{adj}

= adjusted avg case,

N

= reported counts,

K_{adj}

= adjusted closed counts,

P_{adj}

= adjusted paid

On-level premium factor from cumulative rate levels

\text{OLF}_y = \frac{RL_{cur}}{RL_y},\ \text{OLEP}_y = EP_y \times \text{OLF}_y

, RL = cumulative rate-level index, cur = current year, y = accident year, EP = earned premium, OLEP = on-level EP

Disposal (claim settlement) rate at a given maturity

d = \dfrac{C}{U}

, d = disposal rate, C = closed claim counts at the maturity, U = ultimate (or reported) claim counts

Implied average unpaid per open-plus-IBNR claim

\text{Avg Unpaid} = \frac{\text{Case O/S} + \text{IBNR}}{\text{Open Counts} + \text{Unreported Counts}}

, Case O/S = case outstanding reserve, IBNR = incurred-but-not-reported reserve

Roll-forward reserve identity with prior-year development

R_{current} = R_{prior} - P_{period} + \Delta U

, R = held reserve, P = paid during period, ΔU = new ultimate − prior ultimate (development)

Volume-weighted age-to-age development factor

f_{k} = \frac{\sum_{i} C_{i,\,k+1}}{\sum_{i} C_{i,\,k}}

f_k

= factor from maturity k to k+1,

C_{i,k}

= losses for accident year i at maturity k

Cumulative development factor to ultimate including the tail

\text{CDF}_{\text{ult}} = \left(\prod_{k} f_k\right) \times f_{\text{tail}}

f_k

= observed age-to-age factor at age k,

f_{\text{tail}}

= tail factor from last observed age to ultimate

Surplus impact of a reserve error via reserve-to-surplus leverage

E_s = E_r \times \frac{R}{S}

E_s

= surplus error %,

E_r

= reserve error %, R = carried reserves, S = surplus

Age-to-age link factor on a cumulative triangle

f_{t \to t+1} = \dfrac{C_{t+1}}{C_t}

C_t

= cumulative losses at age t,

C_{t+1}

= cumulative losses at next age; computed only on cumulative data

Average case outstanding diagnostic

\text{Avg Case O/S} = \frac{\text{Reported} - \text{Paid}}{\text{Open Counts}}

, Reported = cumulative reported losses, Paid = cumulative paid losses, Open Counts = open claim counts

Ceded loss on an excess-of-loss layer

L_{ceded} = \min\big(\max(L_{gross} - R,\, 0),\; \ell\big)

L_{gross}

= gross loss, R = retention (attachment),

\ell

= layer limit

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